HG 1628 
G62 
1888 
Dopy 1 



LIBRARY OF CONGRESS 



in 



i' 



021 060 447 9 



REVISED AND ENLARGED EDITION, 

SH0RT METH0BS 

FOR 

COMPUTING 

Interest! Diseeant 

100 DAYS INTEREST METHOD. 

Simple Interest. Discount, Compound Interest, 



vT 



nr 

HENRY GOLDMAN, 

AUTHOR OF 

THE ARITHMETICAL DETECTOR. ' ' 
^/STEM FOR DETECTING ERRORS IN TRIAL BALANCES." 
NEW METHOD FOR AVERAGING ACCOUNTS ' ' 
DISCOUNT CALCULATOR. •" "THE EXPERT CALCULATOR,' 
ETC. . ETC. 



-COf\ 



tff 1318R8 

CHICAGO, 1888. 
on 



HC\icU 



COPYRIGHT, 1884 AND 1888, 

BY HENRY GOLDMAN. 

ALL RIGHTS RESERVED. 



LC Control Number 




tmp96 027240 



Jprefaee. 



fl CCUKACY and speed in computing interest 
l\ and discount are important accomplishments; 
/ ~». that they are so rarely found proves to a great 
extent the deficiency of the methods hitherto 
introduced. 
The author offers in this work a method, pro- 
need by experts as the shortest and simplest 
An, which, according to his own judgment, leaves 
iiing to be desired. It can be easily acquired, and 
en once learned is hardly ever to be forgotten. 
Being applicable to all cases which may possibly 
. it is bound to win its way into general use. 
The only objection against the "100 Days Interest 
hod," that it starts with a division, while the 60 
s rule presents an immediate basis, is very shal- 
. and simply illustrates the poor judgment of those 
most persistently urge it. In the first place, 
the initial division a tendency to reduce the 
whole calculation. Secondly, does the decimal ba- 
irrived at decrease all mental labor to a possible 
i mum. And last, but not least, is the one division 
ie beginning a preventive for the m*my divisions, 
it ;t detriment to the GO days rule, 
the repeated loss of fractions, make it 
rule almost impossible to give an absolutely 
•radically correct answer. These are eon- 
irguments which cannot be effectively con- 

e universal application of the LOO days 

puting interest as his own original ide-i. 

not arrogate to himself the first 

the fact that any principal divided by 






('» gives the interest a1 6 per cent, for L,000 days. He 
simply contends that no work published previous to 
the date of his copyright (February 27th, 1884) con- 
tains a method which makes an equally uniform and 
exclusive use of 100 days as a starting point, though 
since the publication of this, Jus ,k 100 Days Interest 
Method. " and its introduction to thousands of book- 
keepers throughout the United States and Canada, 
some unscrupulous parties have, for obvious reasons, 
shamelessly copied it. 

For bank discount no more satisfactory method 
has ever been devised. Based on a strictly scientific 
principle, it combines all the advantages of the so- 
called table methods without sharing any of their 
faults. The time required for opening a table, look- 
ing for page, etc., is more than sufficient to ascertain 
the result by this new method, saving time and in- 
convenience.* It is simpler in principle and more 
convenient in application than the and 12 per cent. 
rules and equally adapted to the 360 and 305 days 
basis. As another advantage the author claims 
that errors resulting from the neglect of fractions, 
which, in the aggregate of many items, may cause a 
considerable difference, are practically avoided by a 
characteristic reduction, decreasing all variations 
from three to twelve times. The consideration of 
the tens of the cents of the principal, whenever the 
number of days exceeds 100, is another of the many 
good features of this method, while the exclusive 
application of decimals places it at the head of all 
methods previously taught. 

Simplified methods for computing interest in con- 
nection with partial payments, ascertaining the pres- 
ent value of a note and an improved process for find- 
ing the compound interest rank also among the most 
original parts of this work. 

The many advantages derived from adopting the 
methods herein given will convince even the most 
skeptical of their excellence. 

Tm: Authob. 



Page. 

| • » DICTION. i 

INTEREST. 

T< ) COMPUTE THE INTEREST. 
The Time Expressed in Days. 

The Year counted at 360 days. ( .) 

The Year counted at 365 days. 14 

The Time Expressed in Months. 15 

The Time Expressed in Years. 15 

T< ) FIND THE PRINCIPAL. 16 

► FIND THE RATE. 17 

* FIND THE TIME. 17 

PARTIAL PAYMENTS. . 18 

DISCOUNT. 

MERCANTILE DISCOUNT. 

Simple Discount. 19 

Combination Discount. 19 

Discount and Per Cent. 20 

BANK DISCOUNT. 

The Year counted at 360 Days. 21 

The Year counted at 365 Days. 22 

TRUE DISCOUNT, 

To Ascertain the Amount. 22 

To Ascertain the Principal. "23 

COMPOUND INTEREST. 

To Find the Compound Interest. 25 

> the Principal. 28 

Find the Rate. 28 

Find the Time. 28 

Find the Present Value. 28 
Mil ms and Annuities. 

To Find the Amount. 20 

To Find the Annuity. 29 

APPENDIX. 

N AND LOSS. 30 

i r on English Money. 32 



Introduction, 



Interest is a compensation for the use of 
money. Principal — The money on which 
interest is computed. Kate Per Cent. — The 
number of cents paid for using $1.00 during 
one year, if not otherwise specified. Amount 
—The sum of principal and interest. 

Accurate Interest being calculated on 
the basis of 365 days to the year, is one seventy- 
third part smaller than Usual Interest, 
which is computed on the basis of 360 days. 

Partial Payments are payments made on 
account of notes etc, reducing the interest 
due at settlement. 

Discount is an allowance for the payment 
of money before due. Face Value — The 
amount of a bill, note, etc, from which discount 
is deducted. Net Proceeds — The difference 
between face value and discount. 

Discount deducted from the face of a bill, 
invoice, etc., without special reference to time, 
is called Mercantile or Trade Discount, 
while Bank Discount is a deduction from 
the face value of a note, draft, etc., equivalent 
t ( > the interest for the exact time from date 
of discount to due date, including three days 
i>ays of grace allowed by law. 

The principal which, at a given rate and in 



— 8— 

a given time, produces a given amount, repre- 
sents the Present Value of the amount and 
leaves, deducted from the latter, the True 
Discount. "Bank Discount" and "True Dis- 
count" are, therefore, essentially different. 

Interest on principal only is also called 
Simple Interest, to distinguish it from 
Compound Interest, which is interest on 
the sum of a given principal and its accumu- 
lated interest. 

Annuities are equal annual payments on 
account of obligations extending over a limit- 
ed or unlimited period. Premiums, amounts 
paid for insurance of life, property, etc. 

The "100 Days Interest Method," forming 
the most important part of this work, will 
prove its superiority over the 30 or 60 days 
rule in offering a threefold basis — the interest 
for 100 days, 10 days and 1 day — which facili- 
tates the computation of interest for any odd 
number of days and dispenses with the 
awkward subdivision into aliquot parts, re- 
quired by the 6 and 12 per cent, rules. Though 
in a number of special cases the application 
of the above-named rules will recommend 
itself, the general usefulness and time and 
labor saving properties of the 100 days method 
have never been approached. 



Interest, 



TO COMPUTE THE INTEREST. 

PRINCIPAL, RATE AND TIME BEING GIVEN. 

THE TIME EXPRESSED IN DAYS. 

THE YEAR COUNTED AT 360 DAYS. 



100 DAYS INTEREST METHOD. 

Any principal showing its own interest at 

36 per cent, for 1000 clays, the interest for 

100 days, 10 clays or 1 clay can be readily seen 

simply removing the decimal point 1, 2 or 

3 places to the left. 

For instance, the interest on 
1371.43 at 36 per cent, for 1000 days is $371.43 

" 100 " 37.14 

« u 10 « 371 

" " 1 clay is .37 

Taking the interest at 36 per cent, for 100 

days, which cover the ordinary business terms, 

basis, the same divided by 

3 gives the interest for 100 clays at 12 per cent. 



4 


a 


i a a 


9 " 


6 


a 


i a a 


6 " 


- 


a 


It a a 


U " 


9 


it 


c a it 


4 " 


12 


u 


a a a 


3 " 



and by pointing off 1 or 2 places, the interest 
10 days or 1 day at the above rates. 



—10— 

The divisors for the most frequent rates can be re 
membered without difficulty, being the result of 36 
divided by the given rate. 

From these fundamental rates the interest 
at other rates can be conveniently obtained 
by adding or deducting the proper aliquot 
parts, for instance: 

6^ per cent, and 5^ per cent, equal 6 per cent, 
more or less one-twelfth. 

7 per cent, and 5 per cent, equal 6 per cent. 

more or less one-sixth. 
1\ per cent, equals 6 per cent, plus one- 
quarter. 

8 per cent, equals 6 per cent, plus one- 
third. 

The interest at 8 per cent, can also be computed 
by rinding first the interest at 4 per cent, and doub- 
ling the same, which is shorter than computing the 
interest at 6 per cent, and adding one-third. 

Note. — Not to conflict with my copyrights and to 
exhibit a certain degree of originality, it has been at- 
tempted to introduce an interest method based on 
.TO per cent. This rate being too small to give ac- 
curate results, the calculation nad to be carried out 
to a further decimal, which practically coincides 
with the decimal principle of computing interest, 
first presented in this work. 

The most simple and practical methods of 

finding the interest at any given rate for 100 

days, 10 days or 1 day, by pointing off 1, 2 

or 3 places from the principal, are presented 

in condensed form in the following 



11 



TAI5LE. 



RATES. 



cent. 



DIVISORS. 

Principal 
divide 

IS 
12 



PARTS. 



To be dedui 



3}$ 


9 


One-eighth. 


4 


9 




4 1 : 


8 




5 


i 


One-thirty-sixth 


~) 1 o 


6 


One-twelfth. 


6 


6 




,; '-2 


5 


One-tenth.* 


. 


•*) 


One-thirty-sixth. 


" ] ] 


4 


One-sixth. 


- 


1 


One-ninth. 




4 


One-eighteenth. 




4 




10 




One-sixth. 


11 


3 


One-twelfth. 


12 


3 





proximately. 

*luct one-tbirty-sixth, deduct one-sixth of one-sixth. 
one-eighteenth " one-third of one-sixfri. 
one-twelftli " one-half of one-sixth. 

being approximately 0.03. 
0.06, 
relfth 

lication by decimals may be preferred. 

The division of the principal is carried out 
- of the cents. This is a new de- 
'tnre and insures greater accuracy than 
>- of ''give and take." Tw< 
result are pointed off as cents, the re- 
hires represent dollars of inten 
The units of the cents of tin'' 



—12— 

principal can always be omitted, their interest 
being, as a rule, too small to cause any varia- 
tion in the answer. This interest method, 
considering the tens of the cents of the prin- 
cipal, gives more accurate results than 
other short methods, which either omit the 
cents entirely or correct the number of dollars. 
To arrive at the interest for any desired 
number of days, multiply the hundreds, 
tens or units of days by the corresponding 
amount of interest and add these products 
together. 

If the number of days is between 90 and 100, de- 
duct from the interest for 100 days as many times 
the interest for 1 day as there are days less. The in- 
terest for 50 days is evidently one-half of the inter- 
est for 100 days, the interest for 25 days one-quarter 
of the interest for 100 days. The interest for 9 days 
equals the interest for 10 days less the interest for 
1 day, which should be remembered whenever the 
number of days contains a 9 in the unit place. 
Examples. 

1. 8723.19 at 6 per cent, for 93 days : 

6)723-1 9 

12-05 = Interest at G r ; for 100 days. 
12X7 -81 " " " L ,; 



Answer: 811*21 -Interest at G % for 93 days. 
2. 8117.51 at 7 per cent, for 117 days'." 

5)117*51 

_ i f>x6) 2-95 == Interest at 7 1 % for 100 days. 

36 5 

49 8 - " 100 " 

.) 

2-87 = Interest at 7 $ for" 100 days 

0-287x1 -29 " " •' 10 •• 

0*028X7 "20 " u "_ | " 

Answer; $3'36 Interest at 7\ for L17 days. 



—13- 

6.87 at 8 per cent, for 5o clays? 

''••8 7 

•63 Interest at 4', for 100 days. 
1-26 = " " 100 " 



0-126x5 "63 Interest at 8 V for 50 davs. 

Answer: 8 "69 Interest at 8 £ for 55 days. 

The interest for 100 days at 5, 1\ , 8, and 
10 per cent, can also be obtained by dividing 
the principal by the component factors of 360. 

5 per cent., divide by 9 and the result by 8. 



i\ (( a a /'♦ tt a a it 


6 
5 

C 


BxAMrLE : 




8123.49 at 10 per cent, for 98 davs? 

6)123-49 

6) 20-58 

3-43 -Interest at 10 % for 100 davs. 
3-4X2 7= " - " 2 " 





Answer: S 3\36 - Interest at 10 % for 98 days. 

To the following rates and corresponding 
numbers of days only, the "100 Days Interest 
Method" cannot be advantageously applied, 
the interest being found without any calcula- 
tion by simply removing the decimal point of 
the principal two places to the left. 

cent. Days. Per cent. Daye. Percent. Days. Percent. Days. 



720 


•v. 


103* 


614 


55* 


9i, 


38* 


1 360 


4 


90 


t 


51* 


10 


36 


1'., 240 


4', 


80 


1H 


48 


101, 


34* 


180 


5 


72 


8 


45 


11 


33* 


11! 


5», 


65* 


S', 


42* 


in, 


31* 


120 


6 


60 


9 


-10 


12 


30 


Approximate! 


V. 













—14— 

THE YEAR COUNTED AT 365 DAYS. 

Apply the "100 Days Interest Method," 
explained and illustrated in the preceding 
pages, and deduct from the interest obtained 
its seventy-third part, in leap years its sixty- 
first. 

Allowing for everv 810. — of interest, 11 cents, 
5- " 7 " 

u (< Q. u rj u 

1. " 1 cent, 

the one-seventy-third part is approximately obtained, 

ExAMPJiE! 

$341.20 at 13 o per cent, for 19 days? 
8)311-2 

1-26 —Interest at i% % f or 100 d ays 

y 2 2-13— " " " 50 " 
''" ^ = " " " 1 •• 

2-09 ^Interest at l 1 ^ % for 49 days 
_ 1 J* 

Answer: 82-06 

Any principal divided by 73 gives the in- 
terest for 100 days at 5 per cent. Any prin- 
cipal divided by 5 gives the interest for 100 
days, 10 days or one day, at 7 '3 per cent, by 
simply removing the decimal point 1, 2, or 3 
places to the left. 

At the following rates and the correspond- 
ing numbers of days the accurate interest 
can be obtained directly by simply removing 
the decimal point of the principal two pla 
to the left. 



lates. Days. Rates. 


Days. 


Kates. 


Days. 


Rates. 


Days 


1 % t 365. 


. 91*. 


s %. 


52* 


10£ 




2 183- 5 


73 


8 


46*' 


11 




3 122 6 


61* 


9 


41* 


12 


30* 


'Approximately. 













—15- 

THE TIME EXPRESSED IN MONTHS AND DAYS. 

Multiply the number of months by 3, annex 
one cipher and add the given number of days. 
If the time is expressed as an interval between 
two dates, make use of the "Improved Time 
Calculator." The " 100 Days Interest Method" 
is applicable in both cases. 

THE TIME EXPRESSED IN MONTHS. 

If the reduction to days cannot be conve- 
niently accomplished, multiply one-fourth of 
the principal by one-third of the product 
of the rate by the number of months and re- 
move the decimal point 2 places to the left. 
Example: 

S428.— at 6 per cent, for 7 months? 
4)428 
107X14 6X7=42 

428 V,=U 



Ans: 814-98 

THE TIME EXPRESSED IN YEARS, MONTHS 
AND DAYS- 

Compute the interest for the given number 
of years first and find the interest for months 
and days according to the directions given 
under the corresponding title. 

THE TIME EXPRESSED IN YEARS. 

Multiply one-fifth of the principal by one- 
half the product of the rate by the number of 
years, and remove the decimal point 1 place 
to the left. 



—16- 

Examivle : 

$1225. — at 5 per cent, for 4 years? 
Answer : 8 245'— 5x4=20 

Xote. — In this instance the multiplication by 10 
counterbalances the moving of the decimal point. 

TO FIND THE PRINCIPAL. 

INTEREST, RATE AND TIME BEING GIVEN. 
THE TIME EXPRESSED IN DAYS. 
THE YEAR COUNTED AT 360 DAYS. 

Multiply the interest by 4, divide by the 
product of one-ninth the number of days by 
the rate, and remove the decimal point 3 
places to 4he right. 

THE YEAR COUNTED AT 365 DAYS. 

Apply the rule stated above and add to the 
answer its seventy-third part, 

THE TIME EXPRESSED IN MONTHS. 

Multiply the interest by 3, divide by the 
product of one-fourth the number of months 
by the rate, and remove the decimal point 2 
places to the right, 

THE TIME EXPRESSED IN YEARS. 

Multiply the interest by 2, divide by the 

product of one-fifth the number of years by 

the rate, and remove the decimal point 1 place 

to the right. 

AMOUNT, RATE AND TIME BEING GIVEN 

THE TIME EXPRESSED IN DAYS. 

Divide the amount by the by 1000 increased 
product of one-quarter the rate by one-ninth 
of the time, and remove the decimal point 3 
places to the right, 



—17— 

THE TIME EXPRESSED IN MONTHS. 

Divide the amount by the by 100 increased 
product of one-third the rate by one-quarter 
the time, and remove the decimal point 2 places 
to the right. 

THE TIME EXPRESSED IN YEARS. 

Divide the amount by the by 100 increased 
product of one-half the rate by one-fifth the 
time, and remove the decimal point 1 place 
to the right. 

TO FIND THE RATE. 

INTEREST, PRINCIPAL AND TIME 

BEING GIVEN. 

Apply the rules given for finding the prin- 
cipal, substituting the known principal for 
the unknown rate. 

TO FIND THE TIME. 
INTEREST, PRINCIPAL AND RATE 

BEING GIVEN. 
THE TIME TO BE EXPRESSED IN DAYS. 

Multiply the interest by 4, divide by the 
product of one-ninth the principal by the 
rate and remove the decimal point 3 places 
to the right. 

THE TIME TO BE EXPRESSED IN MONTHS. 

Multiply the interest by 3, divide by the 
product of one-quarter the principal by the 
rate and remove the decimal point 2 places to 
the right. 

THE TIME TO BE EXPRESSED IN YEARS. 

Multiply the interest by 2, divide by the 
product of one-fifth the principal by the rate 
and remove the decimal point 1 place to the 
right. 



—18— 
PARTIAL PAYMENTS. 

SHORT METHOD. 

1. Compute the interest at 36* per rent, on 
the given principal and the succeeding bal- 
ances, for the corresponding intervals of time. 

2. Compute then the interest on the 06 per 
cent, interest amounts also at 36 per cent, for 
the corresponding totals of the following in- 
tervals of time. 

3. Divide the sum of the interest amounts 
resulting from the second computation by 6 
and place the quotient under the interest 
amounts previously obtained. 

4. Find the total of these 36 per cent, in- 
terest amounts and reduce the same to the 
required rate. 

Example : 6%. interest on principal and balances. 

Days. 36 $, int. Days. 3b $, int. 

Jan. 1. Principal. 81,000.-47, 847.00 73, 83.20 

Feb. 17. 1st paym't 300. — 

Balance 8 700.— 21, 
Mch.10. 2nd paynvt 400. — 

Balance 8 300.-52. 
May 1. 3rd paynrt 100.— 
Balance 8 200.— 

Interest due 13.— .... 6)78.00 

8 213, Answer. 
This method can also be applied to charges, 
increasing the balances of an account. An 
objection to most methods of computing in- 
terest where payments or charges have been 
made is the compounding of interest witliin 
one year. 









14 


14.70 


52, 




.74 
03 


15.00 








.70. 




..6)4 


.20 



Discount 

MERCANTILE DISCOUNT. 
SIMPLE DISCOUNT. 

Multiply the list price or amount of bill by 
the given rate and remove the decimal point 
2 places to the left; or multiply the discounts 
for 1 and 10 cents, 1, 10, 100, etc., dollars by 
the corresponding figures of the amount from 
which the discount should be deducted. 
Examples: 

i 73.21 less 5 per cent. 814.60 less 15 per cent. 

810.— Discount $1.50 

173.21 4.— " .60 

5 .60 " .09 



Discount: 88.66 814.60, Discount : S2.19 

The net can also be directly obtained by 
deducting the rate of discount from 100 and 
multiplying the given price or amount by the 
difference. 

COMBINATION DISCOUNT. 
Subtract each of the given rates from 100 
and multiply the differences. The product 
represents the net of $1.00 in cents and deci- 
mals. The list price or amount of bill multi- 
plied by this product gives the desired net 
price or net amount. Or consider the pro- 
duct of the rate differences as the net of $1.00 
expressed in cents and decimals, deduct the 
same from 81.00 to arrive at the discount and 
obtain the discount for any given amount by 
multiplying the discounts for 1 and 10 cents, 
1, 10, 100, etc., dollars by the corresponding 
figures of the amount from which the discotwrt 
should be deducted. 



-20- 



Example : $425. 

1(H) 
— 40 



60 



Discount, 40, 20 and 5 percent, 

100 100 

20 5 LOO 

456000* 



X 80 



X 95 



•544 



8400.— Discount : 8217-60 
25. '• 13-60 



8125.— Discount : 8231-20 
"Ciphers in the lowest places need not be considered. 

DISCOUNT AND PER CENT, 
To ascertain the rate per cent, correspond- 
ing to a certain discount, annex two ciphers 
to the given rate and divide by 100 less the 
rate. 

COMPARATIVE DISCOUNT AND PER CENT. TABLE 
showing the equivalent of various rates of Discount 
and the corresponding rates % of the resulting net. 



RATES. 


RATES. 


RATES. 


RATES. 


DISCI. 


Mr el 


Disc't. | Per cT 


Disci, Per c't. 


DSC't. 


Per c't. 


5 


5-26 


25 33-33 


50 100*00 


75 


300-00 


i;i 4 


6-67 


30 42-86 


r)3 122-22 


80 


400-00 


10 11-11 


33J£ 50-00 


60 150-00 


85 


566-67 


12 1 2 ; 14-29 


35 53-85 


62%: 166-67 G 


700-00 


15 17*65 


371 o 60-00 


65 185-71 90 


900-00 


16-, 


20-00 


40 66-67 


66% 200-00 909 


LOOO-OO 


20 


2500 


45 81-82 


70 233-33 05 





To find the gross price or amount which 
after deducting a given discount leaves a cer- 
tain net, divide the latter by 100 less the rate. 



ai— 

BANK DISCOUNT. 
Any note giving its own interest at 36 per 
cent., if the year is figured at 360 clays, or at 
per cent, if the year is figured at 365 days, 
100 days, 10 days or 1 day by removing 
the decimal point 1, 2 or 3 places to the left, 
the interest for any given number of days at 
i >r 36A per cent, can be easily found by mul- 
tiplying the interest on each note for 1, 10 or 
100 days by the units, tens or hundreds of the 
corresponding number of days. These prod- 
ucts added together give the Interest or Bank 
Discount on all notes at 36 or 36^ per cent. 

To obtain the interest at any given rate, 
divide the sum of the products by one of the 
divisors of the table, p. 11, according to the 
rate and deduct the part which the correspond- 
ing column indicates, or compute first the 
interest at any of the fundamental rates and 
add or deduct the proper aliquot parts. 
NOTES. 
The dollars of any principal show the interest in 
cents for 10 days at 36 or 36 L 2 per cent.; ten times the 
interest for 10 days equals the interest for 100 days, 
and one-tenth of it the interest for 1 day. 

In computing interest, the fractions of cents should 

be taken in consideration; for instance, the interest 

$345.63 at 36 or 36H per cent, respectively, is for 

days. 634.56; 10 days, 83.46; 1 day, 35 cents. 

The tens of the cents of the principal are consid- 

; whenever the time exceeds 100 days, which se- 

- the greatest accuracy for this method. 



-22— 

Example: Required the Bank Discount on the fol- 
lowing notes, rate, 7 % ? 

„• co ai 
b ^ >* 

eg ca ce 
»h©© 

rH O 



34 

7 

2 

8 



s 3.40 
' J 2.04 

\ 1.44 
/ .65 

\ 3.88 

/ .52 

7.5 " 80 u 7.00 



0.— for 16 days 

2.1 8 " 29 " . 
9.3 " 34 " . 



4.03 
0.2 9 " : 112 " -] .40 



h 

5)23.44 

4.69 

— one-thirty-sixth, .13 

Answer : $4.56 

If the year is counted at 305 days deduct 
the one-seventy-third part from the answer 
arrived at by the above method. 

TRUE DISCOUNT, 

To ascertain the amount which, after de- 
ducting the Interest or Bank Discount at a 
given rate and for a given time, leaves a 
given principal or present value. 

THE TIME EXPRESSED IN DAYS. 

Multiply one-sixth of the rate by one-sixth 
of the number of days, remove the decimal 
point three places to the left, subtract this 
product from 1 unit, and divide the given 
principal by the difference; 



—23— 

; ' en< Value. S1000. . 6 per cent, 93 days. 
Amount? 

ok; 6)93 

Txl 0*015-5 0-9845) 1000.00($1015'74:, Ans. 

THE TIME EXPRESSED IN MONTHS. 

Multiply one quarter of the rate by one- 
third of the number of months, remove the 
decimal point two places to the left, subtract 
the product from 1 unit and divide the given 
principal by the difference. 

Present Value, 8500. — , 8 per cent.. 9 months. Am'nt? 
1)8 3)9 

2 X 3, 1— 0-06=0-94)500(8531-92, Answer. 

THE TIME EXPRESSED (N YEARS. 

Multiply the rate by the number of years, 
remove the decimal point two places to the 
left, subtract this product from 1 unit and di- 
vide the given principal by the difference. 

Present Value 8100.—. 7 per cent., 4 years. Am't? 
7X4. 1—0-28=0-72)100 ($138-89, Answer. 

To ascertain the principal or present value 
which, at a given rate and in a given time, 
produces a given amount. 

Apply the rules given for finding the amount 
with the deviation of adding the product to 1 
unit instead of subtracting it therefrom. 

Amount, 81500. — , 9 per cent. 48 days. 
Present Value? 
0(48 
1-5x8,1+0-012 1-012)1500.00($1482-21, Answer. 

Xote. The most convenient rate for these calcu- 
lations is 6 per cent., as the division by 6 reduces it 
to 1 and saves its consideration. 



-24— 



The amount or present value can also be 
found by dividing the given number of days 
by one of the divisors of the Table, page 11, 
according to the rate, deducting the corres- 
ponding part, removing the decimal point 3 
places to the left, subtracting or adding the 
difference from or to 1 unit, and dividing the 
given principal or amount by this difference 
or sum. The "True Discount" is obtained by 
deducting the present value from the amount, 

TABLE 

of amounts leaving 81. after deducting the discount 
at 4, 5, 6, 7 and 8 % for 33, 63 and 93 days. 



Days 


4*. 


5& 


ft*. 


7%. 


M. 


33 


1.00368 


1.00460 


1.00553 


1.00646 


1.00739 


63 


1.00705 


1.00883 


1.01061 


1.01240 


1.01420 


93 


1.01043 


1.01309 


1.01571 


1.01842 


1.02110 



TABLE 

of present values producing SI. bv adding the interest 
at 4, 5, 6, 7 and 8 %. for 33, 63 and 93 days. 

8C 



Days 


«. 


5*. 


ft*. 


7*. 


33 


0.99635 


0.99544 


0.99453 


0.99362 


63 


0.99305 


0.99ia3 


0.98961 


0.98790 


93 


0.98977 


0.98725 


0.98474 


0.98224 



0.99272 
0.98619 
0.97975 



To ascertain the amount or present value 
leaving or producing a certain present value 
or amount, multiply the corresponding num- 
bers in the above tables by the required 
present value or amount. 



Compound Interest. 

TO FIND THE COMPOUND INTEREST. 

PRINCIPAL, RATE ANDTIME BEING GIVEN. 

The compound interest of Si. 00 can be ob- 
tained by multiplying successively the rate, 
squared rate, cubed rate, etc., by the factors 
of the following table, according to the num- 
ber of years, placing each following product 
2 places to the right under the one preceding, 
and finding their total. 

COMPOUND INTEREST FACTORS. 




6 15 






20 



7 21 35 



8 8 -2s 56 



10 



10 



36 81 



o 
15 



35 



To 



126 



X 

1 
7 



56 28 



M 36 



45 120 210 252 210 120 



- 
- 

X 






4") lo 



X 



26 



Note,- These factors, being the members of pro- 
gressions which stand in simple relations to each 
other, are easily remembered and can be readily ex 
tended to any given number of years. For practical 
purposes it is sufficient to proceed only to the 3rd 
power of the rate. 

Example : 
Compound Interest on $1.00 for 1 years at 5 
5X4=20. . 
25X6= 150. . 
125X4= 500. . 
625X1— 625 

$0-21550625= Answer. 

TABLE 

showing the compound interest of 81.00 
at 1, 5, 6, 7 and 8 £, from 1 to 20 years. 



Years. 


ii 


i 


H 


8* 


l 


0.01000 1 0.05000 


0.06000 


0.07000 


0.08000 


2 


0.08160 | 0.10250 


0.12360 


0.11190 


0.16610 


3 


0.12186 | 0.15763 


0.19102 


0.22501 


0.25971 


1 


0.16986 0.21551 


0.26218 


0.31080 


0.36019 


5 


0.21665 


0.27628 


0.33823 


0.40255 


0.46933 


6 


0.26532 


0.31010 


0.11852 


0.50073 


0.58687 


7 


0.31593 


0.10710 


0.50363 


0.60578 


0.71382 


8 


0.36857 


0.17716 


0.59385 


0.71819 


0.85093 


9 


0.12331 


0.55133 


0.68918 


0.83816 


0.99900 


10 


0.48021 


0.62889 


0.79085 


0.96715 


1.15893 


11 


0.53915 0.71031 


0.89830 


1.10185 


1.33161 


12 


0.60103 0.79586 


1.01220 


1.25219 


1.51817 


13 


0.66507 0.88565 


1.13293 


1.10985 


1.719(52 


11 


0.731(58 0.97993 


1.20090 


1.57853 


1.93719 


15 


0.80091 1.07893 


L.39656 


1.75903 


2.17217 


16 


0.87298 1.18287 


1.51035 


1.95216 


2.12591 


17 


0.91790 J. 29202 


1.69277 


2.15882 


2.70002 


18 


1.02582 1.10662 


L.85434 


2.37993 


2.99602 


19 


1.10685 1.52695 


2.02560 


2.61653 


3.31570 


20 


: 1.19112 


1.65330 


2.20711 


2.86968 


3.66096 



•27- 

y ( Compound [nteresl Table « lit" 
in exhibiting the Compound Inter- 
without the Principal, and not, as usual, Princi- 
pound Interest. It enables to arrive at 
?1 directly, without subtraction. 

To find the Compound Interest with the aid 

of this table whenever the number of years 

■eeds '20, increase the compound interest 

the complementary numbers of years by 

1, multiply these sums and deduct 1 from 
product. The short method for multi- 
ing decimals, explained in the "Expert 
•ulator," can be used to advantage. 



for 28 years? 



Example : 








Compound Interest 


of 


$1.— at 




3-86968 

2 70878 

3870 


X 


1-71819 




39 








M 







- 1 5-64884, Answer. 
To ascertain the compound interest on any 
en principal, multiply the same by the 
compound interest of SI. 00. 

To find, approximately, the number of years 
in which any principal doubles itself through 
imufated compound interest, divide 72 
by the given rate. 

upound interest, if computed semi-an- 
nually or quarterly, is equivalent to the an- 
il compound interest for twice or four 
he number of years at one-half or one- 
quarter the rate. 



—28— 

TO FINDTHE PRINCIPAL. 

COMPOUND INTEREST, RATE AND TIME 
BEING GIVEN. 

Divide the given compound interest by the 

compound interest of $1.00 at the given rate 
and for the given number of years. 

TO FIND THE RATE. 

COMPOUND INTEREST, PRINCIPAL AND 
TIME BEING GIVEN. 

Ascertain the compound interest of $1.00 
by dividing the given compound interest by 
the given principal, increase the same by 1 
and extract the root which the number of 
years indicates. Deduct 1 from the result, 
the difference representing the desired rate. 
Whenever the number of years is such that 
square and cube roots cannot be applied, the 
use of logarithms is indispensable. 

TO FIND THE TIME. 

COMPOUND INTEREST, PRINCIPAL AND 
RATE BEING GIVEN. 

Ascertain the compound interest of $1.00 
by dividing the given compound interest by 
the given principal, — increase the same by 1 
and divide this sum successively by 100 plus 
the given rate, until the quotient equals unity. 
The number of divisions indicates the time 
in years. 

TO FIND THE PRESENT VALUE- 

AMOUNT, RATE AN DTI ME BEING GIVEN. 

Divide the amount by $1.00 plus the com- 
pound interest at the given rate and for the 
given time. 



—29— 

PREMIUMS AND ANNUITIES, 

To find the amount which an annual pre- 
mium of $1.00 will accumulate, divide the 
compound interest at the given rate and for 
the given number of years by the rate, and 
remove the decimal point two places to the 
right. 

! IMPLE : 

Annual Investment, 8 1.00. 5 per cent, i years. 
Accumulated Amount? 
►•2155 ( 81-31, Answer. 

To ascertain the present value of the total 
annual payments of 81.00 during a certain 
number of years apply the above rule, and 
divide the result by §1.00 plus its compound 
interest at the given rate and for the given 
time. 

Example : 
Annual payment $1.00, G per cent., 10 years. 
Present value? 

6 ) 0-7900 (13'18,r7909U3' 1800(8 7-30, Answer, 
To find the annuity with which to pay $1.00 
in a given number of years, divide the rate by 
the compound interest, add the rate to the 
quotient, and remove the decimal point two 
places to the left. 
Example : 

Debt, $1.00. .1 per cent. 1 rears. Annuity? 
8-2155)5-00000(23-2+5 8 -28-2, Answer. 

The results obtained for $1.00 can be ap- 
plied to any amount by multiplying the for- 
mer by the latter. 



/\ppeiAdix. 

GAIN AND LOSS, 

Some important questions frequently oc- 
curring in business find their practical solu- 
tion in the following: 

To ascertain the rate of the gain or loss, if 
the cost price is taken as base, annex two 
ciphers to the difference between the cost and 
selling price and divide by the given cost 
price; or divide the selling price by the given 
cost, multiply the quotient by 100 and de- 
duct 100 from this product. 

To ascertain the rate of the gain or loss, if 
the selling price is taken as base, annex two 
ciphers to the difference between the cost and 
selling price and divide by the given selling 
price; or divide the cost by the given selling- 
price, multiply the quotient by 100 and de- 
duct this product from 100. 

Note. If the selling price is larger than the 
cost the answer is a gain. If the cost is larger than 
the selling price the answer is a loss. The gain 
on the cost price can reach any number of per 
cent., while the gain based on the selling price 
can never exceed 100& the cost in this case being re- 
duced to 0. 



-31— 





increase the cost price: 








t. divide the same 


by 20 and m 


ultiply the quotient by 21. 




• k 


4w 


10 


" 




11. 








8 


" 




0. 








20 


" 




28. 








6 


" 




7. 








10 


M 




12. 




" 




1 


11 




5. 








10 


" 




13. 






** 


3 


11 




1. 






•■ 


8 


" 




11. 








10 


11 




14. 








10 


'• 




kl t 15. 








10 


" 




16. 






• k 


8 


■• 




13. 








3 


" 




" 5. 








10 


•• 




17. 








4 


" 




7. 








10 


11 




18. 








- 


" 




15. 


" 




10 


" 




19. 


To increase the price per piece: 








5 per cent 


subtract 1-8 from the 


price per dozen, 


10 


" 


1-12 




u 


" 


12% 




1-16 




" 


•• 


15 




1-21 
1-36 




.. 


.. 


20 












25 


add 


1-21 to the price 


per dozen. 






1-12 
1-9 




n 


.. 


M 




1-8 




•• 


" 


in 




1-7 
1-6 

1-1 




:: 


" 


•• 




1-3 








" 




1-2 




" 




and 


remove 


the decimal point 


one 


place to the left. 




—32- 

INTEREST ON ENGUl 

Reduce the shillings and pence to 'decimate of one 
pound, and proceed according to the 100 days In- 
terest Method," with the deviation that one more dec- 
imal should be figured. The decimals of the answer 
must be resolved again into s. and d. 

Example : £17, is. 5rf, for 93 days at G % 

6)17-22. 

0*287 Interest for 100 days at 6 %. 
7 tk 6* 



20 



1 
73 



0-267, Interest for 03 days at ( 
1, The year counted at 365 days. 



£0*263 Answer : £ — , os. 3(7. 
NOTES. 
1. To reduce any given number of shillings and 
pence to decimals: Multiply the n^ber of shiU 
lings by5, annex one cipher, and add 1 1-6* times the 
number of pence. 1, 2 or 3 farthings equal M,V % or 
% pence. 

For Instance: 

£13. 18s, 7r/. = 13. 

18X5 = -000 
7X110- 28 

12 



£13-9292. 
| This can be easily figured mentally. 1 
2 To resolve any decimal fraction of one pound 
into shillings and pence : Divide the first ami second 
decimal, considered as a number, by o annex the 
third and fourth decimal to the remainder, and di- 
vide bv 1 l-6t. The result of the first division is the 
number of shillings ; of the second the number dt 
pence. In case the second division l should gne the 
answer 12, increase the number of shillings by 1. 



t Dividing 1118 \ by 4 gives an B PP roximately correct answer * 



